Lemma for Euclid's Elements, Book 7, proposition 2. The original
mentions the smaller measure being 'continually subtracted' from the
larger. Many authors interpret this phrase as mod . Here,
just one subtraction step is proved to preserve the . The
function
will be used in other proofs for iterated subtraction.
(Contributed by Jeff Hoffman, 17-Jun-2008.)
(Proof modification is discouraged.) (New usage is discouraged.)

18.11 Mathbox for Wolf
Lammen

Most of the theorems in the section "Logical implication" are about
handling chains of implications: . With
respect to chains, an rich set of rules clarify

- how to swap antecedents (com12, ...);

- how to drop antecedents (ax-mp, pm2.43, ...);

- how to add antecedents (a1i, ...)

- how to replace an antecedent (syl, ...);

- how to replace a consequent (ax-mp, syl, ...);

- what is, when an antecedent equals the consequent (ax-1, id, ...).

In all these cases, the operands of the chain have no inner structure, or
it is of no importance. These chains are called "simple" here.

There is less support, when the operands are structured themselves. Some
kinds of inner structure involving the operator are best handled by
the symmetric operators and .
But a nested, simple chain
has no such convenient replacement. I can focus on antecedents here,
since a consequent representing a chain is, in conjunction with its
antecedents, just an extended simple chain again.

The following theorems show, how operations on nested chains appear
somehow mirrored: The minor premises of the syllogisms look reverted, in
comparison to their normal counterparts, and while adding an antecedent to
a chain via a1i10 is easy, in nested chains they can be easily
dropped.

Dropping a nested antecedent. This theorem is one of two reversions of
ja153. Since ja153 is reversible, one can conclude, that
a nested (chain
of) implication(s) is just a packed notation of two or more theorems/
hypotheses with a common consequent. (Contributed by Wolf Lammen,
20-Sep-2013.) (Proof modification is discouraged.)
(New usage is discouraged.)

Dropping a nested consequent. This theorem is one of two reversions of
ja153. Since ja153 is reversible, one can conclude, that
a nested (chain
of) implication(s) is just a packed notation of two or more theorems/
hypotheses with a common consequent. (Contributed by Wolf Lammen,
4-Oct-2013.) (Proof modification is discouraged.)
(New usage is discouraged.)

A true wff can always be added as a nested antecedent to an antecedent.
Note: this theorem is intuitionistically valid (see wl-adnestantALT24316)
(Contributed by Wolf Lammen, 4-Oct-2013.)
(Proof modification is discouraged.) (New usage is discouraged.)

Deduction version of wl-adnestant24315. Generalization of a2i12,
imim12i53, imim1i54 and imim2i13, which can be proved by specializing
its hypotheses, and some trivial rearrangements. This theorem clarifies
in a more general way, under what conditions a wff may be introduced as
a nested antecedent to an antecedent. Note: this theorem is
intuitionistically valid (see wl-adnestantdALTOLD24318). (Contributed by
Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.)
(New usage is discouraged.)

Proof of wl-adnestantd24317 not based on ax-37.
(Contributed by Wolf
Lammen, 4-Oct-2013.) (Moved to embantd50 in main set.mm and may be
deleted by mathbox owner, WL. --NM 14-Jan-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)

Theorem *4.86 of [WhiteheadRussell]
p. 122. Place this (and the following
theorems) after bitr1. [ +22] (Contributed by NM, 3-Jan-2005.)
(Proof modification is discouraged.) (New usage is discouraged.)

Deduction adding a biconditional to the left in an equivalence. Move
after bibi1d. [ -25] (Contributed by NM, 5-Aug-1993.) (Proof shortened
by Wolf Lammen, 19-May-2013.) (Proof modification is discouraged.)
(New usage is discouraged.)

In the following notices "experimental" means I have not yet
sufficiently used a definition to be sure it is correct. Anyway I'm
not the owner of the definition and you can use it as you wish if
you think it is correct or replace it by a definition of your own
if you think it is not.

Propositional Linear temporal logic (LTL) is a kind of modal logic. It is
composed of the axioms of classical logic plus the axioms ax-ltl124384,
ax-ltl224385, ax-ltl324386, ax-ltl4 , ax-lmp24388, and ax-nmp24389. In classical
logic, propositions don't depend on the time. In LTL the "world"
evolves. We will imagine the world as a sequence of states with a first
state and future states. Instead of state I will also use the term
"step"
to emphasize that LTL is used to formalize the evolution of process in a
computer. A proposition that is true in one state of the "world"
may be
false in the next one. The proposition means is true in
every state of the world, in the first state as well as in the future
states. It is read " is always true " or " always holds ".
The proposition
means is true
in the next state of the
world. The proposition means that is true in one state
of the world at least but we don't know exactly which one. It can be the
first state, it can be a future state. It is read " is eventually
true " or " eventually holds". When no operator is used in front
of a proposition, it means that is unconditionnaly true or that it
is true in the current state ( depending on the context).
means is
true in every state of the world until is true.